1. Barrier Coverage With Wireless Sensors Santosh Kumar, Ten H. Lai, Anish Arora The Ohio State University Presented at Mobicom 2005

2. Barrier Coverage USA

3. Belt Region

4. Two special belt regions Rectangular: Donut-shaped:

5. How to define a belt region? Parallel curves Region between two parallel curves

6. Crossing Paths  A crossing path is a path that crosses the complete width of the belt region. Crossing paths Not crossing paths

7. k-Covered A crossing path is said to be k-covered if it intersects the sensing disks of at least k sensors. 3-covered 1-covered 0-covered

8. k-Barrier Covered A belt region is k-barrier covered if all crossing paths are k-covered. 1-barrier covered Not barrier covered

9. Barrier vs. Blanket Coverage Barrier coverage  Every crossing path is k-covered Blanket coverage  Every point is covered (or k-covered) Blanket coverage Barrier coverage 1-barrier covered but not 1-blanket covered

10. Question 1 Given a belt region deployed with sensors  Is it k-barrier covered? Is it 4-barrier covered?

11. Reduced to k-connectivity problem Given a sensor network over a belt region Construct a coverage graph G(V, E)  V: sensor nodes, plus two dummy nodes L, R  E: edge (u,v) if their sensing disks overlap Region is k-barrier covered iff L and R are k-connected in G. L R

12. Be Careful! Assumption: If D 1 ∩ D 2 ≠ Φ, then (D 1 U D 2 ) ∩ B is connected.

13. Global algorithm for testing k- barrier coverage Given a sensor network Construct a coverage graph Using existing algorithms  To test k-connectivity between two nodes Question: what about donut-shaped regions? Question: can it be done locally?

14. Is it k-barrier covered? Still an open problem for donut-shaped regions.

15. Is it k-barrier covered? Cannot be determined locally k-barrier covered iff k red sensors exist In contrast, it can be locally determined if a region is not k-blanket covered.

16. Question 2 Assuming sensors can be placed at desired locations  What is the minimum number of sensors to achieve k-barrier coverage?  k x L / (2R) sensors, deployed in k rows

17. Question 3 If sensors are deployed randomly  How many sensors are needed to achieve k-barrier coverage with high probability (whp)? Desired are  A sufficient condition to achieve barrier coverage whp  A sufficient condition for non-barrier coverage whp  Gap between the two conditions should be as small as possible

18. Conjecture: critical condition for k- barrier coverage whp If, then k-barrier covered whp If, non-k-barrier covered whp s 1/s1/s Expected # of sensors in the r-neighborhood of path r

19. k-barrier covered whp  lim Pr( belt region is k-barrier covered ) = 1 not (k-barrier covered whp)  lim Pr( belt region is k-barrier covered ) < 1 non-k-barrier covered whp  lim Pr( belt region is not k-barrier covered ) = 1  lim Pr( belt region is k-barrier covered ) = 0

20. L(p) = all crossing paths congruent to p p p

21. Weak Barrier Coverage A belt region is k-barrier covered whp if lim Pr(all crossing paths are k-covered) = 1 or lim Pr( crossing paths p, L(p) is k-covered ) = 1 A belt region is weakly k-barrier covered whp if crossing paths p, lim Pr( L(p) is k-covered ) = 1

22. Conjecture: critical condition for k- barrier coverage If, then k-barrier covered whp If, not k-barrier covered whp What if the limit equals 1? weakly weak

23. Determining #Sensors to Deploy Given:  Length (l), Width (w), Sensing Range (R), and Coverage Degree (k), To determine # sensors (n) to deploy, compute  s 2 = l/w  r = (R/w)*(1/s)  Compute the minimum value of n such that 2nr/s ≥ log(n) + (k-1) log log(n) + √log log(n) s 1/s

24. Simulations Using this formula to determine n,  The n randomly deployed sensors provide weak k-barrier coverage with probability ≥0.99.  They also provide k-barrier coverage with probability close to 0.99.

25. Summary Barrier coverage Basic results Open problems  Blanket coverage: extensively studied  Barrier coverage: further research needed